Math 341

Side-Angle-Side on the Sphere

          Recall the following fact about planar triangles:
          (SAS)  If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.
          This is Euclidís Proposition 4.

        Decide whether or not SAS is true on the sphere.  If you think that it is true on the sphere, then give a convincing argument that shows that it is true.  If you think that it isnít true, explain why it is false, show how to modify the statement to make it true on the sphere, and then prove your modified statement.  In either case, explain and justify any decisions that you had to make in order to come to your conclusion.  Try to write explanations that would convince a reasonable skeptic.  You will probably find it helpful to thoroughly understand Euclidís planar proof before trying to decide what happens on the sphere.